The azimuth calculator above gives the direction from one point on the Earth to another, measured clockwise from north in degrees. It computes that direction two ways — on the WGS 84 ellipsoid using Vincenty’s inverse formula, and on a sphere using the standard great-circle expression — and prints both, together with the difference between them, so you can see the size of the model error rather than take one number on trust.
Arb Digital publishes both numbers because the answers genuinely differ, and how much they differ depends on the route. Between the Greenwich Observatory and Manhattan the two models disagree by about four hundredths of a degree and by fifteen kilometres of distance. Between Moscow and Buenos Aires the azimuth gap widens to nearly two tenths of a degree. Neither figure is large enough to matter for a walk, and both are large enough to matter for a survey.
What This Azimuth Calculator Does
It returns the initial azimuth from your start point to your end point, referenced to true north. It also gives the same figure computed on a sphere, the back azimuth — the direction you would set off in to return, which is almost never the forward bearing plus 180 — the magnetic azimuth once you supply your local declination, and the geodesic distance.
The summary line reports the difference between the ellipsoidal and spherical answers explicitly, in degrees for the bearing and in your chosen unit for the distance, plus the compass point and the equivalent in mils for anyone working with military or artillery conventions.
This page is about direction. If the distance is what you are after, our great circle distance calculator is built around it and gives the haversine answer in several units. If your coordinates need converting first, use the coordinates converter, and if you want the point exactly opposite your start, the antipode calculator covers that special case.
How to Use It
- Enter the start coordinate in decimal degrees. This is where you are standing or where the instrument sits.
- Enter the destination coordinate. Order matters: azimuth is directional, and swapping the two points does not simply add 180 degrees.
- Look up your magnetic declination for the start point and today’s date, and enter it with east positive. Leave it at zero if you only need true bearings.
- Choose a distance unit. Nautical miles are the convention for marine and air navigation.
- Compare the two azimuths in the summary line. If the gap is smaller than your instrument’s precision, the model choice does not matter for your job.
The Formula and Which Model It Uses
On a sphere the initial azimuth from point 1 to point 2 is:
θ = atan2( sinΔλ · cosφ₂ , cosφ₁ · sinφ₂ − sinφ₁ · cosφ₂ · cosΔλ ), normalised into 0–360°
That is the great-circle bearing, and it treats the Earth as a perfect sphere. The Earth is not one: it is flattened by about one part in 298, so the polar radius is roughly 21 km less than the equatorial radius. The ellipsoidal answer on this page comes from Vincenty’s inverse method on the WGS 84 reference ellipsoid, with semi-major axis 6,378,137 m and flattening 1/298.257223563 — the same ellipsoid GPS receivers report positions against. Vincenty iterates to a tolerance of 10−12 radians, which is sub-millimetre at these scales.
How much do they differ? Measured with this tool: Greenwich to New York gives 288.431° spherical against 288.470° ellipsoidal, a gap of 0.039°, with distances of 5,579.6 km and 5,594.6 km, a gap of 15.0 km. Tokyo to Sydney gives a bearing gap of 0.052° and a distance gap of 33.7 km the other way. Moscow to Buenos Aires gives 0.177° and 16.6 km. As a rule of thumb the bearing disagreement stays inside about two tenths of a degree while the distance disagreement can reach a few tenths of a percent, and both grow with route length and with how much latitude the route crosses.
Practically: 0.05° is about 1.3 metres of lateral error over 1.5 km, which no handheld compass will resolve. It is also several metres of error on a survey baseline, which is exactly why national geodetic agencies work on the ellipsoid. NOAA’s National Geodetic Survey distributes its INVERSE and FORWARD geodetic programs for this computation, and they remain the reference implementation to check against.
Why the Back Azimuth Is Not the Forward Plus 180
On a flat map, reversing a bearing means adding 180 degrees. On a globe it does not, and the error can be large. A great circle crosses each meridian at a different angle, so the direction of the same route measured at the far end is generally not the reciprocal of the direction measured at the start.
The defaults on this page make the point sharply. Greenwich to New York sets off at about 288°, roughly west-north-west. New York to Greenwich sets off at about 51°, roughly north-east. The reciprocal of 288 is 108, which is south-east, and would take you towards the Sahara. The discrepancy of more than 50 degrees is not an error in either number; it is what a great circle does across that much longitude.
The gap shrinks to nothing on short routes and on routes that run along the equator or along a single meridian. It grows with the longitude span and with latitude. Any navigation that assumes reciprocal bearings over long distances is going to be wrong, which is one reason great-circle routes are flown as a series of legs with the heading updated at each waypoint.
True North, Magnetic North and Grid North
There are three norths and they rarely agree. True north points at the geographic pole and is what this calculator computes by default. Magnetic north points at wherever the Earth’s magnetic field currently leads a compass needle, which is a different place, moves measurably every year, and differs from true north by an angle called declination that depends on where you are standing.
Declination can be tens of degrees. It is close to zero along certain lines and exceeds twenty degrees in parts of Alaska, Siberia and the far south. Applying the wrong sign is a classic mistake: the convention used here is that magnetic azimuth = true azimuth − declination, with declination positive to the east. A declination of 10° E means the compass reads 10 degrees less than true.
This tool deliberately publishes no declination figures. They change continuously and any table typed into a web page is wrong within a year or two. Get the current value for your exact location and date from NOAA’s World Magnetic Model, which is the model your phone and your chartplotter are almost certainly using, and paste it into the field above.
Grid north is the third one: the direction of the northward grid lines on a projected map such as UTM or a national grid. It differs from true north by a small angle called convergence that grows away from the central meridian of the zone. Survey and military work has to be explicit about which of the three a bearing refers to, because confusing them silently is how a route ends up a kilometre off.
Azimuth, Bearing and the Words People Mix Up
Azimuth is measured clockwise from north through the full 360 degrees, so due east is 090 and due west is 270. That is the convention used throughout this page. What surveyors call a quadrant bearing is written differently: N 45° E, S 30° W, always as an angle of 90 degrees or less measured away from north or south. The same direction expressed both ways looks like two different numbers.
Artillery and some military mapping use mils rather than degrees. The NATO convention divides the circle into 6,400 mils, so one degree is 17.78 mils; the true mathematical milliradian would give 6,283 to a circle, and the two are not interchangeable. The summary line gives the NATO figure and labels it as such.
Astronomy uses azimuth too, for the horizontal direction of a celestial object, and there the convention is not universal: some older references measure azimuth from south rather than north. Always check the datum a source is using before comparing two azimuths from different fields.
Arb Digital builds free tools that name the model, cite the source and publish the error rather than hiding it. Browse the library, or tell us what your readers keep asking.
Browse Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Reversing a bearing by adding 180. That works on a plane and fails on a globe. Read the back azimuth from the tool instead.
- Getting the declination sign backwards. East declination is positive here, and magnetic azimuth is the true azimuth minus it.
- Using a declination value from an old chart. Declination drifts every year. Look up the current value for your location and date.
- Holding the initial bearing for the whole route. A great-circle heading changes continuously; keeping it fixed gives a rhumb line, which is a different and longer path.
- Mixing degrees-minutes-seconds into a decimal field. Convert first, or the answer will be silently and badly wrong.
Related Free Tools From Arb Digital
Measure the route itself with the great circle distance calculator, convert coordinate formats with the coordinates converter, find the opposite point on Earth with the antipode calculator, work in Cartesian space with the 3D coordinate converter or the 3D distance calculator, and switch to polar form with the polar coordinates converter. Everything else is in the free online tools hub.
Frequently Asked Questions
It is a horizontal direction measured clockwise from north, in degrees from 0 to 360. Due east is 090, due south is 180 and due west is 270. This calculator gives the azimuth from your start point towards your destination.
Yes, and this tool prints both. Measured on the routes used as examples on this page the bearing difference ranged from about 0.04 to 0.18 degrees and the distance difference from 15 to 34 kilometres. It grows with route length and with the latitude crossed.
Because a great circle crosses each meridian at a different angle. From Greenwich to New York the initial bearing is about 288 degrees, while from New York to Greenwich it is about 51 degrees, not 108. The gap grows with the span of longitude covered.
Subtract the magnetic declination at your location, taking east declination as positive. A true bearing of 100 degrees where declination is 10 degrees east reads as 90 degrees on the compass.
Because declination changes from year to year and any figure published on a web page goes stale. Look up the current value for your exact position and date using NOAA’s World Magnetic Model and enter it here.
Azimuth runs clockwise from north through a full 360 degrees. A quadrant bearing is written as an angle of 90 degrees or less from north or south, such as N 45 degrees E. They describe the same direction in different notation.
No. The bearing along a great circle changes continuously, so a fixed heading traces a rhumb line instead, which is a different and generally longer route. Long-distance navigation updates the heading at waypoints.
This tool implements published geodetic formulas on the WGS 84 ellipsoid for education and general reference. It is not certified for navigation, surveying or any safety-critical use, where an appropriate geodetic package, current magnetic data and a qualified professional are required.
